Booms, bubbles, and crashes (Job Market Paper)

Transcription

1 Booms, bubbles, and crashes (Job Market Paper) Yue Shen Queen s University January 16, 014 Abstract In this paper we attempt to answer the fundamental question of why bubbles exist and design a tentative tax/subsidy scheme that reduces bubbles. We first generate bubbles from a simplified version of Abreu and Brunnermeier (003) model, where rational traders optimally ride the bubble for sufficient rise in price. We then show that the model is equivalent to a reverse discriminatory price (first-price) common value auction. We further reveal that bubbles exist because in these auctions prices fail to reflect asset values, and two (reverse) bid shading incentives together give rise to the bubbles. A surprising implication is that a tax on the capital gain during the boom, which is designed to suppress the bubble, may actually inflate the bubble. A simple tax/subsidy schedule is devised which can reduce the size of bubbles by half. In an extended model, we analyze how traders decide to first purchase and then sell. When the price is so high that no one wants to buy any more, not even the trader with the highest signal, the bubble bursts. In a unique equilibrium, traders collectively create and optimally ride the bubble for sufficient long. Bubble size is decreasing in the transaction cost, and traders maintain a substantial markup between their purchase and sale prices. I am grateful to James Bergin and Frank Milne for their patience and numerous valuable comments. I thank Ruqu Wang for his insightful comments and suggestions. I have also been benefited from conversations with Marco Cozzi, Rui Gao, Steve Kivinen, Karl Skogstad, Marie-Louise Vierø, Tianyi Wang, Jan Zabojnik, Mark Zhou, and participants in the department seminars. All errors are my own. Department of Economics, Queen s University, Kingston, Ontario, Canada. sheny@econ.queensu.ca

2 1 Introduction The asset bubbles that are relevant in this paper are broadly defined as a relatively large upward price deviations from their fundamental values, which can last for extended periods. Such phenomena have long been intriguing to economists because they not only affect the financial sector, but also have huge impact on the real economy. Historical examples of bubbles include the Dutch tulip mania of the 1630s, the South Sea bubble of 170 in England, the Mississippi bubble in France, the Great Recession of 199 in the United States, DotCom bubble in the late 1990s, and the most recent housing bubble crash since 008. However, bubbles have been difficult to explain and generate theoretically. One major hurdle is the no trade theorems. One version of the theorem states that, if, the initial allocation is efficient, there is common knowledge that all traders are rational, and agents have common priors about the distribution of asset values and private information, then agents do not have incentives to trade 1. In a dynamic setting, in particular, the standard neoclassical theory precludes the existence of bubbles by backward induction. Facing these impossibilities, economists elicit trade by relaxing different aspects of the assumptions in the theorems. For surveys, refer to Brunnermeier (009), Brunnermeier (001), Brunnermeier and Oehmke (01) and Scherbina (013). One strand of the literature allows heterogeneous priors and agents to agree to disagree. Harrison and Kreps (1978) show that when agents disagree about the probability distributions of dividend streams, a trader may buy an asset at a price that exceeds her own valuation because she believes that in the future she will find a buyer with more optimistic belief. In a similar model, Scheinkman and Xiong (003) justify this reasoning by studying overconfidence as a source of disagreement. In a related strand of literature, Allen et al. (1993) and Conlon (004) allow agents to have heterogeneous priors and hold worthless assets in hopes of selling it to greater fools. In stead of relying on agreeing to disagree, it is the lack of common knowledge about the distribution of belief or a higher order belief about asset value that generate the bubbles. One advantage of these models is that all agents are rational. Both of the above strands help explain in theory why trade exists in the market, but they are hard to apply to the bubbles in the real world because, to some extent, they lack a clear price path with steadily growth and then a sudden collapse. Moreover, the complexity of these models implies that agents need to be hyper-rational to deduce the posterior beliefs 1 See Grossman and Stiglitz (1980) and Milgrom and Stokey (198).

3 so that bubbles emerge. These restrictions have prompted many economists to be inclined to believe that behavioral traders must be active in the market. Recent empirical and experimental studies and psychological studies have provided convincing evidence that agent s behavior is far from rational in daily trade. Hence, the third strand of literature studies bubbles that arise from the interaction between rational traders and behavioral traders. In De Long et al. (1990), rational traders buy the asset and induce positive-feedback traders to follow and buy the asset from (behavioral) passive traders. After the price is further pushed up, the rational traders can profitably unload. Although simple and intuitive, the bubble in their model arises exactly because behavioral traders are buying. If we remove the positive-feedback traders and leave only the passive traders, there will be no bubble. In addition, the bubble exists also because time is discrete in their model. If we transform the model into a continuous time one and assume that not all traders can receive the full pre-crash price if they all sell together, then the bubble will not survive because rational traders will then have incentive to preempt each other, which suppresses the size of the bubble until it drops to zero. The non-robustness in the above model actually confirms the efficient markets hypothesis, which states that prices are consistent with the fundamental values, and that well-informed sophisticated investors will undo the price impact of behavioral traders so that any bubble cannot persist for an extended period. Herding is an intuitive explanation of the rise of bubbles. Literature on rational herding challenges the efficient markets hypothesis and attempts to simulate the emergence of bubbles and subsequent crashes 3. Avery and Zemsky (1998) introduce a sequential trade model, where a trader is informed with probability µ and a liquidity trader with probability 1 µ. To produce an informational cascade as well as a crash, three dimensions of uncertainty are needed: the signal about the asset value, the quality of the signals, and the uncertainty about the µ. The market maker adjusts the price after observing each round of transaction. If many of the poorly informed traders are herding and trading with the market maker, the price will keep increasing above its liquidation value, and then, after many liquidity trades, the bubble bursts. Lee (1998) studies an information avalanche, where a trader must pay a transaction cost but can decide when to trade. The market maker can adjusts the price only ex post every round. All traders are informed but differ in the precision of their information. A partial informational cascade can occur if many use the wait and see strategy but be shattered by an extreme signal that triggers all See Shiller and Akerlof (009) and Shefrin (008). 3 See Brunnermeier (001) and Chamley (004) for surveys on this topic. 3

4 previous inactive traders to sell. However, both models suffer from the critique that the bubble followed by a crash happens with very small probabilities. In addition, in almost all rational herding models, traders have essentially only one opportunity to trade. Abreu and Brunnermeier (003) (henceforth AB003) also challenge the efficient markets perspective. In their model, the price keeps growing as a result of behavioral traders actions, and, at some random moment, the growth rate of fundamental value falls behind that of the price, hence a bubble emerges. Rational traders become aware of this sequentially. The bubble will burst when a certain fraction of traders have sold. A trader s problem is to decide when to sell the asset. Ideally one wants to sell just prior the crash. But this is difficult. A trader understands that, by selling early, she can make a small profit; by selling late she might be able capture a large price appreciation but also have the risk of getting caught in the crash. She needs to balance between selling at a higher price (riding the bubble) and avoiding getting caught in the crash. A key ingredient is that traders are uncertain of their position in this awareness queue, i.e., a trader does not know how many others became aware ahead of her. Therefore, every trader has a different posterior belief about the time at which the bubble first emerges and the bubble size. This dispersion of belief induces dispersion of exit strategies, which is what allow the bubble to arise and grow. AB003 captures the greater fool dynamic well. More importantly, they show that it is optimal for rational traders to ride the bubble, which is shown to be consistent with recent empirical studies of stock market data. Temin and Voth (004) show that a major investor in the South Sea bubble knew that a bubble was in progress and nonetheless invested in the stock and hence was riding the bubble profitably. Brunnermeier and Nagel (004) and Griffin et al. (011) both study the Tech bubble in the late 1990s. They show that, instead of correcting the price bubble, hedge funds turned out to be the most aggressive investors. They profited in the upturn, and unloaded their positions before the downturn. Doblas-Madrid (01) (henceforth DM01) constructs a discrete-time version of AB003 and addresses certain issues that were under criticism in AB003. In AB003, the exogenous price increase because of behavioral agents are buying, and only the arbitrageur s liquidation decision is explicitly modeled. In addition, the price does not respond to rational traders sales until a certain threshold is crossed. DM01 removes the behavioral agents from the model and allows rational traders to buy and sell every period, hence price is (partially) determined in equilibrium and becomes responsive to sales. Two complications arise out of this change. First, when the price is low no one wants to sell so there is no trade. To initiate sales at low prices, traders are randomly drawn and hit by liquidity 4

5 shocks and must sell all their shares. Second, to maintain an increasing price path, it is necessary that the money available for traders to purchase the asset increases over time, since the expected aggregate forced liquidation is constant over time. We will further discuss these complications in the extended model. Observing the price, agents update their belief and, when there is a large enough price drop, they liquidate. Hence, the price path dynamic is more realistic. AB003, together with DM01, provide a framework of bubbles that is intuitive and justifiable, both theoretically and empirically. However, why bubbles exist is still not clear. AB003 explains that the lack of common knowledge prevents traders from perfectly coordinating, and hence the backward induction has no bite and a bubble can exist. This, however, is only a necessary condition for bubbles to exist. Brunnermeier and Morgan (010), who study a discrete trader version of AB003, show that such a game can be recast as an auction. But they do not pursue this direction. In particular, they are silent on how the emergence of bubbles is related to auction theory. In this paper, we will attempt to answer the fundamental question: why bubbles arise? We first show that a simplified version of AB003 is equivalent to a reverse common value auction. As a result, why bubbles arise is because price fails to reflect the true value of the object in this auction. Then we identify two bid shading incentives in traders strategies that generate bubbles. Next, we design a simple tax/subsidy scheme which can reduce the size of bubble. Specifically, we simplify AB003 by removing all non-essential features, and transform the uncertainty on the dimension of time in AB003 to the dimension of value/price. That is, in addition to a common prior belief about the asset s fundamental value, each trader receives a private signal, and their strategies are functions of price only. This transformation abstracts time away from the model and removes the stark assumption of sequential awareness. As a consequence, the intuition of AB003 becomes clearer and, with price being the only concern for the agents, it is not difficult to show that the model is equivalent to a reverse (procurement) auction with a common value outside option. The auction is one where traders simultaneously submit bids, and the lowest bids win and each receives their individual bid amount, while the rest receive a unknown common value outside option, which corresponds to the fundamental value of the asset. The results of the benchmark model, as well as models in the recent auction literature, show that the prices 4 do not converge to the true value in these common value auctions 5. It thus becomes clear that why bubbles arise is for the same reason why prices deviate 4 The price here is the marginal bid, i.e., the highest winning bid or lowest losing bid. 5 A detailed review of this literature is postponed to Section... 5

6 from the true values in common value auctions. As a trading mechanism, common value auctions fail to aggregate information dispersed in the bidders and bidders overly revise their bids. Equivalently, market often overacts to shocks because of the inability to reach a consensus among the privately informed investors, and, as a result, each investor optimally rides the bubble. This equivalence might be interpreted as bad news for efficient market hypotheses, because it re-confirms that the bubbles and crashes arise from the information asymmetry. Nevertheless, it prompts further studies in this issue from a mechanism design perspective to develop policies that could possibly minimize bubbles and economic fluctuations. It is worth mentioning that three frictions are at work allowing the bubble to arise and grow: 1. limited short selling,. lack of common knowledge of the population s belief, and 3. unobservability of individual s trade. We further show that, in this discriminatory price reverse common value auction, two (reverse) bid shading 6 incentives contribute to the rise of bubbles: 1. traders efforts to avoid both (reverse) winner s and loser s curse and. traders price-setting incentives (because they get what they bid upon winning) in first-price auctions compared to pricetaking behavior in second-price auctions. The result is that the bubble size equals price setting incentive. Compared with previous auction literature, the two incentives in my model can be explicitly separated in the bidding strategies, which may also help explain why price deviate from true value in the discriminatory price common value auctions. In addition, they have opposite response to the change of ratio between winning slots and bidders, which can explain different results from several models. Based on above interpretation, several tax schemes are tested and a surprising result is that that taxing only on pre-crash sales and not otherwise actually makes the bubble size larger. This result puts an alert to the policies that some governments use to subdue the housing prices. With the above lessons learned, we design a simple tax/subsidy scheme that could reduce the size of the bubble by half. Targeting at the uniform price common value auction where there is no bubble, if we subsidize the pre-crash sales and let them all have receive the marginal bid, there will be no bubble. However, the budget is unbalanced. A revised scheme which subsidize early sellers and tax on late sellers, and which results in identical payoff for pre-crash sellers and a balanced budget, reduces the size bubble by half. In the extended model, we analyze how traders decide to first purchase and then sell, with the hope of riding the bubble and selling before the crash. The price keeps increasing exactly because there are rational traders buying. Traders buy at a low price and wish to 6 Bid shading is the practice of a bidder placing a bid below what she believes a good is worth. 6

7 sell at a high price before the crash. With the transaction cost, they will no longer seek to buy if the current price is too close to their scheduled sale price, otherwise they will not be able to profit. In the unique equilibrium, a bubble exists when transaction cost is small compared to the dispersion of belief about fundamental value, and bubble size decreases in transaction cost. In addition, a small transaction cost induces a large gap between the the purchase and sale prices. In contrast to AB003, where an exogenous burst threshold is imposed, the bubble bursts in our model because the price is so high that no one, not even the trader with the highest signal, wants to buy; hence, the price stops increasing, and it becomes common knowledge that the asset is indeed overpriced. When the transaction cost is large compared to the dispersion of beliefs about the fundamental value, there is no bubble. A trader stops buying when the price approaches her estimate of fundamental and sells after uncertainty is resolved. There is no bubble riding. Both AB003 and DM01 require that the price increases exponentially with time so that simple solution forms are possible. Our extended model allows the price to increase in an arbitrary way, because time is not essential in our model. Instead, what matters is the price, and traders decisions only depend on the price. The remainder of the paper is organized as follows. Section investigates the origin of bubbles by examining the benchmark model. Section.1 introduces the benchmark model and characterizes the equilibrium. Section..1 shows its equivalence to the common value auction. Section.. shows that prices generally do not converge to true value in a common value auctions. Section..3 clarifies the two reverse bid shading incentives which induce the bubble, Section..4 devises a simple tax/subsidy scheme to reduce the bubble size, and Section..5 shows the two incentives respond differently to κ. In Section 3, we study the extended model. Section 3.1 shows how traders strategy spaces are reduced. In Section 3., we solve an individual trader s problem backwards, by first solving a trader s optimal selling price and then her stop-buy price. The equilibrium is characterized in Section 3.3. Section 3.4 briefly discusses a recession, which is the reverse process of the bubble. Section 4 concludes. A benchmark model: the origin of bubble.1 The benchmark model In this section we present a simple model that is very close to AB003 and then illustrate its equivalence to a discriminatory price common value auction, which shows that bubbles 7

8 occur for the same reason as in the common value auctions, where information aggregation fails and price deviates from true value in the limit. To focus on the intuition the origin of bubbles, I will make the benchmark model as simple as possible. For a detailed explanation and justification of the assumptions, please see AB003. Consider a model that parallels AB003. Specifically, in an environment with continuous time and only one asset, the asset s fundamental value is θ, which is a random variable and unobservable. There is a unit mass of risk neutral rational traders (henceforth traders), each holding 1 unit of asset at the beginning. Without loss of generality, each trader s asset position is normalized to [0, 1], so that only limited short selling is allowed. A trader can sell her shares at any time. The price can be publicly observed and is denoted p(t). When t > 0, the price starts to increase continuously and deterministically 7. At any time, when the price rises above θ, we say there is a bubble. As in AB003, the backdrop is that the asset price keeps increasing continuously and deterministically, which can be interpreted as behavioral agents buying the asset. These behavioral agents believe that a positive technology shock has permanently raised the productivity and growth rate of this industry, and they simply keep buying this asset, which pushes up the price, as was the case of the tech bubble in the 1990s. The belief of behavioral agents is also confirmed by the fact that rational traders are all holding the asset. However, as will be introduced shortly, each rational trader has a private belief on θ, and they will start to sell one by one when they gradually believe that the price is too high. Initially this sale is disguised by the noise in the price, but when more and more rational traders have sold, its impact on the price will be observed by all behavioral traders, which ultimately subverts their belief, and all of them start to sell. Although there is no price noise in our model, the above story is summarized as a threshold: we assume that when a fraction of κ traders has sold, the price stops increasing and jumps instantly to its fundamental value and stays there thereafter, i.e., the bubble bursts. In line with AB003, we call this an endogenous crash. Note that behavioral agents are not modeled here, and with the threshold assumption, we essentially remove behavioral traders from our model and a rational trader is only concerned about how many other rational traders have sold. An example of the price path is depicted in Figure 1. The bubble size is the gap between the price p(t) and θ when p(t) > θ. There is an upper bound B for the bubble size, and the bubble will also burst when the bubble size is larger than B. We call this exogenous crash. This is to rule out the possibility that all traders hold the asset forever and never 7 In contrast to AB003, we no longer require that price increases exponentially and no discounting is necessary. 8

9 price bubble size θ t bubble bursts Figure 1: An example of the price path sell. After all, when an asset s value is more than the GDP of the whole economy, no one, not even behavioral traders, will still believe that the price is a fair reflection of its fundamental value. We are only interested in the endogenous crash, so we assume that B is large enough so that the bubble will always burst due to threshold κ. θ has density φ(θ) distribution over [0, ). We restrict to two alternative distributions: an improper uniform distribution on [0, ) 8 and an exponential distribution with density φ(θ) = λe λθ9. They both give simple solutions, but we will focus on the uniform case in most of our analysis because of its simplicity. At t = 0, each trader receives one, and only one, private signal s, which is uniformly distributed on [θ η, θ + η ]10. s can be regarded as a trader s type. Since θ is random, a trader is not sure about her signal s position in [θ η, θ + η ], i.e., a trader does not know how many others signals are lower than hers, and how many are higher than hers. This is an important element in the model because this lack of common knowledge prevents agents from perfectly coordinating with each other. In contrast, in the standard finance literature, perfect coordination leads to backward induction, which rules out the possibility of a bubble. Note that in the extended model, we will allow traders to buy first then sell, and the price increases precisely because 8 The improper uniform distribution on [0, ) has well defined posterior belief when we specify how signals are distributed. 9 The uniform prior case is adapted from Li and Milne (01) and the exponential prior case is adapted from Abreu and Brunnermeier (003) 10 When θ is close to zero, some traders will receive negative signals, which has no consequence since traders know that θ cannot be negative. The implication that some low signal traders posterior has smaller support will be addressed shortly. 9

10 of rational traders purchases. Let Φ(θ s) be the posterior CDF about θ of a trader with signal s, and φ(θ s) be the corresponding PDF. In the exponential prior case, Φ(θ s) = eλη e λ(s+ η θ), and in the e λη 1 uniform prior case, Φ(θ s) = θ (s η ). Given signal s, the support of the posterior η is [s η, s + η ]11. Figure depicts the posterior belief about θ for trader s, s and s. These specifications make all agents posterior belief have exactly the same shape, except φ(θ s) φ(θ s ) φ(θ s ) s s s θ Legend: φ(θ s) φ(θ s ) φ(θ s ) Figure : Posterior beliefs a horizontal shift. Given one s belief, a trader will sell at the price where her marginal benefit and cost of selling at a slightly higher price are equal. Since no one knows how many others are below or above herself, everyone behaves in a similar way. If the chance of being a low type trader is large enough, then, when observing that the price is increasing, it is optimal for everyone to ride the bubble sufficiently high before selling. We assume that a trader s purchase and sale cannot be observed by other traders. There is no discounting. Corollary.1. A trader will not hold the asset forever. To simplify the problem and focus on the results and interpretations, we assume the following: Assumption.1. A trader uses a trigger strategy: she sells only once, whereby she sells all her shares and will never buy back. 11 When s is below η, the support of posterior of trader s is truncated at 0 because θ cannot be below zero. This cause traders in the lower boundary [ η, η + ηκ] behave differently from those above. This will be clarified in Proposition.1 where we characterize the equilibrium. The rest of the analysis will ignore this special case. 10

11 In this reduced strategy space, a trader need only consider at which price to sell. AB003 starts with weaker assumptions and derives this trigger strategy as a result. In this benchmark model, we simply assume the trigger strategy. In the extended model, we will start with weak assumptions and derive similar results instead of assuming them. Assume all other agents use strategy P (s), where s is signal. To further simplify the analysis, we have the following assumption. Assumption.. P ( ) is continuous, strictly increasing and differentiable on ( η + ηκ, ) 1. This guarantees that agents with higher signals must sell at higher prices. It also implies that P 1 ( ) is well defined. Lemma.1. Any equilibrium strategy P ( ) must be that P (s) s is bounded, s. See Appendix A.1 for proof. Let burst price be p T, hence p T = P (θ η + ηκ) and θ = P 1 (p T ) + η ηκ. Since θ is a random variable, so is p T. Suppose an agent i decides to sell at price p, then if p < p T, she will be able to flee the market before the crash; otherwise, she will get caught. By inverting this relationship, we know that she will get caught in the crash if θ [s η, P 1 (p) + η ηκ], and she will flee the market successfully before the crash if θ [P 1 (p) + η ηκ, s + η ]. Therefore, given that all others use strategy P (s), the expected payoff for a particular trader s is E[R s] =max p P 1 (p)+ η ηκ θ=s η θφ(θ s)dθ + s+ η P 1 (p)+ η ηκ pφ(θ s)dθ (1) The above formulation is a static game. We first solve for the static equilibrium strategy P (s). Given that all other use this strategy, it is not difficult to verify that a trader s problem is well defined (SOC< 0). Then we show that the strategies derived from the static formulation are the same as those from the original dynamic game, hence traders do not change their initial plan even when they update beliefs as price increases. Lastly, we claim, in Proposition.1, that this is a unique equilibrium and clarify traders strategies at the lower boundary (when s is close to η ). Differentiating E[R s] w.r.t p, imposing P (s) = p, and set de[r s] dp 1 See footnote 11 1 = = 0, we have [ P (s + η ] 1 ηκ) φ(s + η ηκ s) P 1 Φ(s + η ηκ s) () 11

12 The FOC (Equation ()) can be interpreted in terms of marginal benefit (MB) and marginal cost (MC). For a trader who evaluates selling at p vs. p +, the benefit of selling at p + instead of p is. The cost is that she could get caught in the crash if that happens in between p and p +. This equals loss p θ (due to bubble burst) multiplied by Φ(P 1 (p + ) + η ηκ s) Φ(P 1 (p) + η ηκ s) 1 Φ(P 1 (p) + η ηκ s) (the probability of bursting between p and p + ). Dividing both sides by and letting 0, we have MB= 1 and MC=(p θ) 1 φ(s + η ηκ s) P 1 Φ(s + η which is the LHS and RHS of the FOC. ηκ s), Proposition.1. There is a unique equilibrium, in which a trader sells at price B, if her signal s < η + ηκ P s (s) = s + η ηκ + B, if her signal s η + ηκ This gives rise to a bubble of the size B, where B = { ηκ in the uniform prior case 1 e ληκ λ in the exponential prior case See Appendix A. for proof 13. The equilibrium strategy is depicted in Figure 3. p s + η ηκ + B P s (s) B η η + ηκ s Figure 3: Equilibrium strategy Then we show that the strategies derived from the static formulation are the same as those from the original dynamic game even when they update beliefs as price increases. In 13 The proof of uniqueness requires an additional technical assumption: for those traders in the lower boundary [ η, η + ηκ], when any positive mass of traders sell at the same price and the bubble bursts right at that price, only some of them (random draw) can sell at pre-crash price. 1

13 Equation (1) the trader s belief is Φ(θ s) with support [s η, s + η ], which is her initial belief. See Figure 4 panel (a) (where an arbitrary density φ(θ s) is depicted). However, as the price increases, this belief can change. Consider that a trader re-examines her situation at any price p e > 0. Given that all others use equilibrium strategy P (s), the bubble will burst at p T = P (θ η +ηκ) and hence θ = P 1 (p T )+ η ηκ. When the current price p e is such that P 1 (p e )+ η ηκ < s η, where s η is the lowest possible θ, bubble bursting is impossible for trader s, i.e., when p e < s η +ηκ the bubble will certainly not burst. When price p e increases such that p e > s η + ηκ, from trader s s point of view, the bubble can burst at any moment. The fact that the bubble has not burst below p e shrinks the support of θ from below and hence changes the belief of trader s. Specifically, that the bubble has not burst below p e implies that θ cannot be below θ = P 1 (p e ) + η ηκ (i.e., the bubble bursts right at price p e ). Hence, the support of θ is now [P 1 (p e ) + η ηκ, s + η ], and the updated belief about θ is Φ(θ s, p e ) = Φ(θ s) Φ(P 1 (p e)+ η ηκ) s), with corresponding density φ(θ s, p e ) = φ(θ s) 1 Φ(P 1 (p e)+ η 1 Φ(P 1 (p e)+ η ηκ) s) ηκ) s). See Figure 4 panel (b). φ(θ s) s η P 1 (p e )+ η ηκ s + η θ (a) Original belief φ(θ s) φ(θ s, p e ) s η P 1 (p e )+ η ηκ s + η θ (b) New belief φ(θ s, pe) Figure 4: Update of belief The trader s s problem now is P 1 (p)+ η ηκ s+ η E[R s, p e ] =max θφ(θ s, p e )dθ + pφ(θ s, p e )dθ (3) p P 1 (p e)+ η ηκ P 1 (p)+ η ηκ 13

14 Notice that φ(θ s, p e ) differs from φ(θ s) only on its denominator, which is constant (does not include θ). Therefore problem (3) and problem (1) are equivalent and their solutions are identical, and this result does not depend on particular forms of prior distributions of θ or signal distributions. This equivalence is similar to the equivalence between first-price sealed bid auction and Dutch auction. In the Dutch auction, a bidder has a initial posterior belief about the highest signal among others. When price is decreasing and no one has claimed the object yet, the upper bound of this belief also decreases. However their biding price does not change under the new belief and, as has been showed many times in the literature, Dutch auction is strategically equivalent to the first-price sealed bid auction. This is because, in Dutch auction, the new belief differs from old one only by a constant denominator, which does not change the optimal choice of bidding price. The difference is that, the new belief in our benchmark model takes the form of auction takes the form of φ( s) Φ(constant s) φ( s), while the new belief in Dutch 1 Φ(constant s). If we further consider the equivalence between a reverse first-price sealed bid auction 14 and reverse Dutch auction 15, the new belief in the φ( s) reverse Dutch auction is exactly of the form, and the lower bound of posterior 1 Φ(constant s) belief is increasing but the hazard rate at the planned selling point does not change.. Why do bubbles exist As mentioned in the introduction, why bubbles exist is not clear in AB003. AB003 explains that the lack of common knowledge prevents traders from perfectly coordinating, and hence the backward induction has no bite and a bubble can exist. However, this is only a necessary condition. Brunnermeier and Morgan (010), who study a discrete trader version of AB003, show that such a game can be recast as an auction. But they do not pursue this direction. In particular, they are silent on how the emergence of bubbles is related to auction theory. In the rest of this section, we will show that the benchmark model is equivalent to a reverse common value auction, and bubbles exist because in this auction the price fails to reflect asset value. We then show that the equilibrium selling price P (s) can be decomposed as: The marginal bidder incentive is traders effort to avoid both winner s and loser s curse; the price setting incentive is traders effort to set the price in first-price auc- 14 Reverse auctions will be introduced in Section..1. Simply put, Reverse auctions are procurement auction, where the auctioneer wants to buy and bidders want to sell. Winners are those who bid the lowest. 15 reverse Dutch auction can be thought as an English auction, but with only the increasing price observable. Bidders drop outs are unobservable. 14

15 P (s) = s + marginal bidder incentive + price setting incentive = s + η ηκ + 1 Φ(s + η ηκ s) φ(s + η ηκ s) tions, compared to the price-taking behavior in second-price auctions. Also, the size of the bubble B is B =price setting incentive = 1 Φ(s + η ηκ s) φ(s + η ηκ s) Therefore, it is the two incentive that generate the bubble. This result sheds light on how to reduce bubble size, which we discussed briefly. In addition, the fact that the two effects have opposite response to a change of κ can explain different results among different models...1 Relationship to an auction The benchmark model in Equation (1) is actually a pure common value auction (mineral rights model), except that bidders are continuous instead of discrete. Specifically, this is a reverse discriminatory price (first-price) sealed-bid multi-unit auction with a single unit demand and common value outside option. We will explain the terms step by step. All traders are selling/bidding in this one single auction. See Figure 5. The solid line is the price price θ marginal bid κ 1 κ t Figure 5: Equilibrium strategy path. Winners in the auction are those who sell before the crash, whose selling/bidding prices are represented by the thick solid line. The losers are those who are caught in the crash, and the thick dashed line represents their planned selling prices, though never happened since bubble bursts before they have a chance to sell. All traders/biders are 15

16 involved in this single auction. The fact that there is a fraction κ of winners means there is the same mass of continuous identical contracts are up for tender, and each trader/bidder bids to procure only one contract. We say contracts because this is a reverse auction (also called procurement auction), where the auctioneer wants to buy, say, labor or service via a mass of κ of contracts, and traders/bidders are selling their labor or service to the auctioneer. Bidders who bid lowest (selling at lowest prices) win, and each winner is awarded a contract and receives contract value equal to their bidding price (selling price). In contrast, in the normal auctions, winner are those who bid highest. It is called a discriminatory price auction because winner pays/gets her own bidding price, which corresponds to the first-price in the single object auction. The winners need not exert any effort in the contract. The losers receive a common value outside option θ. An agent s beliefs about θ is the same as what was introduced in the benchmark model. The expected payoff with a signal s is exactly the same as in Equation (1). To win, a trader necessarily bids low enough to be in the low range. But the lower the bid, the lower the payoff upon winning. Since an agent does exert any effort and either receives θ or p (her bidding price), so she compares the two alternatives and her bidding strategy only depends on her belief about the common value outside option θ: the lower her belief about θ, the lower she bids. This is why we say it is a common value auction. The idea is even clearer when we further alter the above auction so that there is no outside option. Instead, winners need to exert an uncertain, common effort θ to fulfill the contract. Then the expected payoff of trader s changes to E[R s] =max p s+ η P 1 (p)+ η ηκ (p θ)φ(θ s)dθ This expected payoff is different from the auction with outside option in Equation (1), but the first order condition turns out to be the same as in Equation (), so the equilibrium strategy is the same. AB003 demonstrates that the lack of common knowledge prevents traders from perfectly coordinating, and hence backward induction has no bite and a bubble can exist. That, however, is only a necessary condition for bubbles to survive, but does not explain why bubbles arise. Still, it is not clear as to why rational traders are willing to wait and sell at prices above fundamental, causing a bubble to arise. AB003 also indicates that the model is not a global game and strategic complementarity is not at work. Brunnermeier and Morgan (010) show that such a game can be recast as an auction. However, they did not pursue this direction further. In particular, they are silent on how the emergence of bubbles is related to auction theory. In what follows, we will show that as a trading mech- 16

17 anism the above auction, as well as some other common value auctions, fails to aggregate information and prices deviate from the asset values... A review of common value auction literature: price convergence Whether this a bubble is equivalent to whether, in the reverse auction, the marginal/pivotal bidding price is higher than or equal to (or lower than) θ (See Figure 5). The marginal/pivotal bid is the highest winner s bid, which is also the lowest loser s bid. We have seen that, in the benchmark model, the marginal bidding price is higher than the fundamental value. This subsection shows that, this result holds in more general settings as demonstrated in the recent auction literature. That price converges to true value of object in the discreet bidder/objects common value auction is called information aggregation. Theories in common value auction have been used to demonstrate the price convergence and market efficiency. Wilson (1977) and Milgrom (1979) showed that, in a first-price common value single unit auction, the winning bid converges in probability to the value of the object as the number of bidders n becomes large. Milgrom (1981) showed that, in a uniform-price (second-price) common value auction with k identical objects for sale and each bidder only desiring one item, where all winning bidders pay the (same) k + 1th bid (which corresponds to the second price in single object auction), if we fix k and let the total number of bidders n, then the price (the highest loser s bid, which is also the k + 1th highest bid) also converges to true value v. These results demonstrate that, while no one knows for sure the exact value, the auction, a price formation process, aggregates diffuse information through the economy and the price tends to reveal the value. All above results not only require the monotone likelihood ratio property (MLRP), but also require that the likelihood ratio approaches to zero. Let f(s θ) be the density distribution of a bidder s estimate when true value is θ. MLRP requires that, if θ 1 < θ, f(s θ 1) f(s θ decreases in s, which most distributions satisfy. But ) the price convergence also requires that f(s θ 1) f(s θ 0 as s s, where s is the upper bound ) of the support of s. However, as recent literature has shown, the requirement that the likelihood ratio 0 turns out to be too strong, and when it does not hold, the price generally fails to converge to true value in the common value auction. Kremer (00) shows that both first- and secondprice single object common value auctions fail to aggregate information. Jackson and Kremer (007) show that, in the discriminatory price common value auction with k identical objects for sale and each bidder desiring only one item, the information aggregation also fails and the price does not converge to asset value even when both k and n. 17

18 In particular, they show that the marginal/pivotal bid in the auction is lower than the true value of the asset. The only situation where the price does converge to true value is Pesendorfer and Swinkels (1997), who show that when both k and n, the price converges to true value in the uniform price common value auctions. To compare the strategies in our benchmark model and standard auctions and gain further insight, it is convenient if we expand and alter the benchmark model (which is a discriminatory price common value auction) to uniform-price and to private value auctions, under the framework of reverse auction with continuous bidders and contracts. Under uniform-price, all winners receive the marginal bidder s bidding price; in the private value auctions, a winner needs to exerts an effort equal her signal s, instead of an common value θ. We will use some of the results from these variant models. These price convergence in normal auctions with both k and n, as well as the benchmark models, are summarized in Table 1. Normal auctions, k, n Benchmark model Uniform price Discriminatory price Uniform price Discriminatory price Common value P V marginal P < V p T = θ p T > θ Table 1: Price convergence In Jackson and Kremer (007), the marginal bidding price is lower than true value. Since the benchmark model is a reverse auction, it is natural that the marginal price is higher than the true value in this reverse auction, which is exactly the case in the benchmark model. This conformity shows that bubbles arise from the benchmark model is not due to some peculiar assumptions. It holds in more general cases. Note that in uniform price cases, there is no bubble, which prompts potential opportunities to reduce the bubble. The equivalence of our benchmark model to the discriminatory price common value auction, and the fact that the marginal bid in this auction does not converge to true value, show that why bubbles arise is for the same reason of why marginal bid does not equal object true value. As a result, to generate bubbles, we do not need such fancy information structures as heterogenous prior beliefs, or explicit higher order beliefs. One as simple as common value auction is enough. In addition, when private information is dispersed in market participants and no one is sure about the asset fundamental value, the market fails to magically aggregate these 18

19 information 16, and price fails to reflect average belief. Riding the bubble becomes optimal to each investor. Compared to previous literature, this interpretation makes the intuition especially clear that the bubbles and crashes arise from the information asymmetry, which is an intrinsic characteristic of the market. This is a bad news for market efficiency hypothesis, and implies that every time a news arrives, there could be a coordination failure and hence a bubble arises. In what follows, we first clarify two incentives that shape the bidding strategies, then show how they affect the marginal bid so that it deviates from true value, and hence bubbles arise...3 Two bid shading incentives We have seen that the benchmark model, as well as recent auction literature show that the marginal price/bid deviate upward (downward) from the true value in the reverse (normal) discriminatory price common value auctions. One may ask, why is it necessarily a upward deviation in the reverse auctions. Why can t it be a downward one in the reverse auctions. That is, why is there necessarily a crash in the benchmark model, instead of a jump-up at the threshold κ? After all, when we defined the endogenous crash, we indicated that the price jumps, but the direction was not specified. A more fundamental question is that, when no one is sure about the fundamental value, why is a bubble and a crash created in equilibrium? Why it is impossible for them to sell such that there is no bubble or there is even a negative bubble where price jumps up at threshold κ? There does not seem to be any particular reason in AB003, or DM01, or our model that forces the traders to sell high enough that a bubble emerges. The fact that the behavioral traders drive price up is actually not critical to inducing a bubble. There is no clear answer in AB003 or DM01 to this question; we will try to answer this question in this subsection by showing that traders have two (reverse) price shading incentives, which can explain why bubble arise and also help explain why marginal bid deviates from true value in the discriminatory price common value auctions. 1. Marginal bidder incentive: bidders to avoid both winner s and loser s curse, so that the marginal bidder bids exactly at true value. It is defined as bidding by conditioning on being the marginal bidder. This incentive exists in all common value auctions and 16 To summarize, three frictions are at work allowing the bubble to arise and grow: 1. limited short selling,. lack of common knowledge of the population s belief, and 3. unobservability of individual s trade. 19

20 in common value auctions only. This incentive is best illustrated in uniform-price common value (normal) auction, where there is no interference of the other incentive. The winner s curse is that, if a bidder turns out to be the winner, it means that her signal s is one of the highest among all bidders and hence biased upward. Hence, if she have bid naively, it is very likely that she overestimated and overbid. This is more striking when there are n bidders but only one object for auction. Hence sophisticated bidders uniform-price common value auction bid conditioning on winning. The loser s curse is that, upon losing, a bidder finds that her signal is among the lowest (biased downward) and hence she may have underestimated and underbid. This is more striking when there are n bidders and n 1 objects for auction, in which case, sophisticated bidders would bid conditioning on losing. In general cases where there are n bidders and k < n objects, the bidding strategy in uniform-price common value auction is to bid conditioning on being the marginal bidder. In fact, in both single object and n 1 objects cases, we have always been conditioning on being the marginal bidder. Bidding conditioning on being the marginal bidder is actually the only symmetric equilibrium biding strategy in uniform-price common value auctions. It is because, if the marginal bidder s bid is higher than true value, then all winners will be paying a price higher than true value due to the uniform-price setting. Then all bidder will try not to win, which is no longer an equilibrium. If the marginal bid is lower than true value, then a bidder has incentive to raise her bid to have higher probability of winning, which essentially has no impact on the price she will pay upon winning. Hence this not an equilibrium either. Ex ante, no one knows whether she will be the marginal bidder. Hence in equilibrium, everyone behaves as if she is the marginal bidder. This is why in the normal discrete bidder uniform-price common value auctions, the equilibrium strategy is E[v X 1 = s, Y k = s], where X 1 is my signal, and Y k is the kth highest signal among all other bidders. In our uniform-price variant of benchmark model, the bidding/selling strategy is s + η ηκ, because conditional on she is the marginal bidder, i.e. her signal is θ η + ηκ, her bid will be exactly the true value θ. Therefore, η ηκ is bidders effort to avoid both winner s and loser s curse and to bid exactly at θ if being marginal bidder.. Price setting incentive: traders to set the price to seize extra surplus in discriminatory price auctions, compared to price-taking behavior in uniform price auctions. 0

21 This incentive exists in all discriminatory price auctions and in discriminatory price auctions only. In uniform price (second-price) auctions, a bidder does not pay what she bids and her bid essentially has no impact on her payment. Hence she behaves like a price-taker. In contrast, in discriminatory price (first-price) auctions, a bidder pays exactly what she bids upon winning. Hence, in normal auctions she has incentive to her bid. It is well-known that, in the first-price private value auctions, if a bidder bids exactly her signal, she always has zero surplus. To extract positive surplus, she shades her bid. In first-price common value auctions, this incentive still exists. But when she lowers her bid, she also lowers her winning probability. So the equilibrium strategy would be to balance these two forces. In the benchmark model, where the lowest bids win the contracts, this incentive is reversed: upon winning, a bidder in discriminatory price gets paid what she bids, so she has incentive to increase her bid. As demonstrated previously, the price setting incentive only appears in the discriminatory price auctions. In Table, the competition effect in the uniform prior case is (positive) ηκ, and in the exponential prior case 1 e ληκ λ (recall that the hazard rates are 1 and ηκ. These terms are actually the inverse of the hazard rates in the FOCs λ 1 e ληκ prior cases, respectively). If we re-write Equation (), we have P = (s + η ηκ) + 1 Φ(s + η ηκ s) φ(s + η ηκ s) in the uniform prior and exponential where we assume P = 1 and ignore it. Note that the FOCs in the uniform price cases do not have the term 1 Φ(s+ η ηκ s). In uniform price reverse auctions, since a φ(s+ η ηκ s) bidder does not pay what she bids, she would bid infinitely high if she is guaranteed to win. When she has to consider the possibility of losing, then the situation where she is the marginal winner matters. In this case, 1 Φ(s+ η ηκ s) is exactly what she φ(s+ η ηκ s) can add to her bid to balance between seizing extra value and not forgoing too much the opportunity of winning. Finally, notice that both ηκ and 1 e ληκ λ increase in κ, which means that in these reverse auctions, higher κ relieves competition and leads to higher bids (selling prices). These two incentives are well-known in auction literature, maybe under different names. But in previous literature they are entangled in the bidding strategies and the definitions are somewhat ambiguous. The benchmark model provides a unique opportunity where we can see them explicitly defined and separated, due to its information structures and 1